The variational form associated with Equation (7) is:

where, is the normal unit vector on the border and, a normal deriva-

tive on this border. To solve Equation (9), one subdivides the domain delimited by x = 0 and x = H₁ according to Ox and y = 0 y = H₂ according to Oy, into

triangular finite elements where N₁ and N₂ are positive integers. One has points for the mesh, as represented in Figure 2(a).

Let us consider a triangular finite element e and its three vertices 1, 2 and 3 with their respective coordinates (x₁, y₁), (x₂, y₂) and (x₃, y₃), as shown in Figure 2(b).

In this element e, can be approximated by a continuous solution hence the approximate solution on whole domain is given by [12] .

m being the number of triangular finite elements of discretization.

For triangular finite elements, can be seeking in the linear form:

After eliminating the constants a, b and c in above Equation (10), one obtains the carrier’s density of finite element e as follows:

where indicate the values of carrier’s density at the vertices (1, 2, 3) and the coefficients given by:

is the surface of triangular element such as:

.

In each element, if one chooses as a trial function, these expressions of and introduced in the Equation (9) leads to the relation:

where and.

The elements of the matrix (Stiffness matrix) are given for each triangular element by the integral [12] :

Those of the mass matrix are determined by:

The elements of the matrix, taking account of the boundary conditions and the orientation of the normal vector, on the borders, are given by:

is a column matrix representing the second member. Its elements are given by:

Taking into account the contribution of each triangular element, one obtains the carriers density U to each node of the mesh after assembling all of the matrices and et. Thus this carrier’s density, after minimization of the variational form (9), obeys to the equation:

or inversely:

A and L are the global matrix of Equation (13). The vector is the unknown excess minority carrier’s density.

By considering that the minority carriers flow in y-direction is weak compared to that in x-direction, the current-voltage characteristic of the solar cell is obtained by determining the photocurrent density J and the photovolatge V calculated from the relations (21) and (22) respectively as follow [1] [3] :

q indicates the electron’s charge and

, represents the thermal voltage, Nb: the doping in the base, , the intrinsic carrier’s concentration.

Using a numerical code that we conceived from this theoretical approach based on the finite element method, we have solved the continuity equation and analyzed the effects of recombination velocity at the grains boundary on the electric parameters of the solar cell. To validate our results, we initially determined the number of finite elements of the grid beyond that the solution converges. Thus, we have plotted in Figure 3, the minority carriers density photogenerated at the y = 0 plane boundary, for various values of the number of finite elements with.

These curves shows that the convergence of the solution is obtained when N ≥ 30 i.e. for 1682 finite elements and 900 points of the mesh.

To ensure a good compromise between the precision and the computing time, we chose to use N = 30 finite elements.

We represented in Figure 4 and Figure 5, the excess minority carriers density according to base depth x, and the grain width y, when the solar cell is respectively illumination on its front surface and its back surface.

One notes that for the front surface illumination, the carriers density increases according to x, reaches its maximum around x = 0.175 mm, then it decreases and cancels when x = 0.3 mm. As for illumination on the back surface of the solar cell, the curve

preserves the same profile but the maximum is reached around x = 0.25 mm.

We can also note a slight variation of these densities according to y, for both illuminations.

The effect of grain boundary recombination velocity on the minority carrier’s density is highlighted in Figures 6-9.

The grain boundary recombination velocity S_gb, reveals the carriers losses by recombination, at the grains boundaries. These losses significantly influence the profile of the

excess minority carriers density as shown in these figures, represented according to the base depth and the grain width, respectively for S_gb = 0, S_gb = 10³ cm/s, S_gb = 10⁴ cm/s and S_gb = 10⁶ cm/s. S_j and S_b being fixed at 10⁶ cm/s and 10⁵ cm/s.

When S_bg = 0, there are no losses by recombination at the grains boundaries. The photogenerated carrier’s density is more important. When S_gb increases, the recombination becomes more important and the carrier’s density decreases. We can note that at the grains boundaries (borders y = 0 and y = H₂), the carriers density is weak. That is more and more remarkable when S_gb increases [1] .

When S_bg reaches the value of S_j = 10⁶ cm/s, the carriers density becomes null at the boundaries.

Figure 10 and Figure 11 below represent the photogenereted minority carriers densities according to the base depth, with a boundary recombination velocity S_gb = 10⁵ cm/s, and the junction recombination velocity S_j, respectively equal to 10⁴ cm/s and 10⁵ cm/s. The effects of S_j are definitely observable on these curves. That permits us to note the

widening of the space charge region Z_CE and the reduction, even the cancellation of the carriers density at y = 0 and y = H₂ planes boundary. These profiles of the carrier’s density permit to highlight the recombination phenomenon which is an important one in the solar cells operation [10] .

The same effects are observed when the solar cell is illuminated on the back surface as illustrated in Figure 12 for the same value of S_bg = 10⁵ cm/s.

The photocurrent density is determined starting from Equation (21). We have repre- sented it in Figure 13, versus the junction recombination velocity S_j, for various values of S_gb.

As waited, when S_j is equal to zero, the photocurrent density is null. When the junction recombination velocity S_j increases, the photocurrent density increases and is saturated when S_j reaches a critical value, corresponding to the short-circuit operation of the solar cell.

When the junction recombination velocity S_j is close to zero, the photocurrent density is null. That corresponds to the open-circuit operation of the solar cell. While when S_j is very large, the photocurrent density is constant and equal to the short-circuit current. We can also note that the open circuit or the short-circuit operation depends to the boundary recombination velocity S_gb. That is due to the fact that when S_gb is higher, losses by recombination are important.

The photovoltage is one of the characteristic elements of the solar cell. Using our numerical code, we have determined the photovoltage and represented it in Figure 14,

according to the junction recombination velocity for various values of S_gb. This figure shows that when the junction recombination velocity S_j is close to zero, the photovoltage is equal to the circuit-open photovoltage Vco. and when S_j increases from zero to approximately S_j = 10² cm/s, the photovoltage remains constant and equal to the open circuit photovoltage. Beyond that value, the photovoltage decreases and cancels out when S_j reaches the short-circuit operation, for a value of S_j approximately equal to 10¹² cm/s. We can also note that the open circuit photovoltage is larger when S_gb is smaller.

The knowledge of the current-voltage characteristic is very important for the solar cell characterization. Using our code of calculations, we have determined and represented in Figure 15, the current-voltage characteristics of the solar cell for various values of the grain boundaries recombination velocity S_gb.

As awaited, these characteristics are in conformity with the solar cell operation. Indeed, one notes that when the photovoltage is equal to zero, the photocurrent density is equal to the short-circuit photocurrent density Jcc. When the photovoltage increases, the photocurrent density decreases and cancels, when the photovoltage reaches the open circuit photovoltage Vco.

One can notice that the open circuit photovoltage Vco and the short-circuit photocurrent density Jcc decrease when the grains boundaries recombination velocity S_gb increases [1] .